# Find all possible values of $p$ and $q$ for which $\cos^{-1}\sqrt p+\cos^{-1}\sqrt{1-p}=\large\frac{3\pi}{4}$

$\begin{array}{1 1}(a)\;p=0,q=\large\frac{1}{2}&(b)\;p \leq 2,q=1\\(c)\;0 \leq p\leq 1,q=\large\frac{1}{2}&(d)\;None\;of\;these\end{array}$

Step 1:
Given :
$\cos^{-1}\sqrt p+\cos^{-1}\sqrt{1-p}+\cos^{-1}\sqrt{1-q}=\large\frac{3\pi}{4}$
$\Rightarrow \cos^{-1}\sqrt p+\sin^{-1}\sqrt p+\cos^{-1}\sqrt{1-q}=\large\frac{3\pi}{4}$
Let $\cos^{-1}\sqrt{1-p}=\theta$
$\cos\theta=\sqrt{1-p}$
$\sin\theta=\sqrt p$
$\sin^{-1}x+\cos^{-1}x=\large\frac{\pi}{2}$